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NUMERIC Mathematics Lecture SeriesNipissing University, North Bay, Ontario
Exploring the Math and Art Connection6 February 2009
Dr. Daniel JarvisMathematics & Visual Arts
Education
Professor Katarin MacLeodMathematics & Physics
Education
Dr. Mark WachowiakComputer Science
6 February 2009 Jarvis MacLeod Wachowiak 2
Workshop Overview Introduction: Exploring the Math/Art Connection Golden Section: Ratio/Proportion in Ancient Greece Activity 1: Creating your own golden section bookmark Tesselations: Transformations in 20th Century Europe Activity 2: Creating your own tessellation pattern Fractals: Iterations in 21st Century Activity 3: Creating your own fractal designs Technology: Simulations from Nature Video Clips: : “Donald Duck In Mathmagicland” (1959)
and “Life by the Numbers” with Danny Glover (2006) Resources: Galleries, Artists, Books, Conferences, and
Stuff Questions and Comments
6 February 2009 Jarvis MacLeod Wachowiak 3
AN INTRODUCTION TO RATIOIn mathematics, a ratio is defined as a comparison of two numbers. A proportion is simply a comparison of two ratios.
Perhaps the most famous mathematical ratio/proportion is what is known as the Golden Section or the “Divine Proportion.”
This proportion is derived from dividing a line segment into two segments with the special property that the ratio of the small segment to the large segment is the same as the ratio of the long segment to the entire line segment.
6 February 2009 Jarvis MacLeod Wachowiak 4
Geometry has two great treasures: One is the Theorem of Pythagoras; the other the division of a line into extreme and mean ratio.
The first we may compare to a measure of gold; the second we may name a precious jewel.
Kepler (1571-1630)
6 February 2009 Jarvis MacLeod Wachowiak 6
HISTORICAL OVERVIEW
THE RENAISSANCE“DE DIVINA PROPORTIONE” (1509)
WRITER: FRA LUCA PACIOLIILLUSTRATOR: LEONARDO DA VINCI
6 February 2009 Jarvis MacLeod Wachowiak 7
HISTORICAL OVERVIEW
THE MODERN ERA:
ARTISTS OF THE 19TH AND 20TH CENTURIES
SEURAT, DALI, MONDRIAN,COLVILLE
6 February 2009 Jarvis MacLeod Wachowiak 8
HISTORICAL OVERVIEW
SONY PLASMAGRAND WEGA$17 000 CANADIAN16:9 ASPECT RATIO(APPROX. 1.78)
APPLE IMAC$2500 CANADIAN36.8/22.8 = 1.614(GOLDEN APPLES?)
6 February 2009 Jarvis MacLeod Wachowiak 9
TEACHING HOW TO FIND THE GOLDEN SECTION
[I] ALGEBRAICALLY
1
11
1 0
1 1 4 1 1
2 1
1 5
2161803398
2
2
2
x
x
xx x
x x
x
x
x
( ) ( )( )
( )
. ...
x 1
= 1.61803 (Phi)NOW, BEGINNING WITH ANY GIVEN LENGTH (L):
NEXT LARGEST SECTION
LENGTH X (1.61803)
NEXT SMALLEST SECTION
LENGTH/(1.61803) OR MORE SIMPLY, LENGTH X (0.61803)
6 February 2009 Jarvis MacLeod Wachowiak 10
TEACHING HOW TO FIND THE GOLDEN SECTION
[II] GEOMETRICALLYBEGIN WITH A SQUARE; EXTEND ONE SIDE
FROM MIDPOINT, CUT AN ARC FROM FAR CORNER TO EXTENDED LINE
COMPLETE RECTANGLE & INTERNAL SQUARES
6 February 2009 Jarvis MacLeod Wachowiak 20
THERE IS GEOMETRY IN THE HUMMING OF THE
STRINGS.
THERE IS MUSIC IN THE SPACING OF
THE SPHERES.PYTHAGORAS
(C.A. 582-500 B.C.)
6 February 2009 Jarvis MacLeod Wachowiak 21
RELATED PHENOMENA
DYNAMIC SYMMETRY: ROOT RECTANGLES IN GREEK DESIGN (AS OPPOSED TO STATIC)
PLATONIC SOLIDS: REGULARITY, RECIPROCITY, & GOLDEN RECTANGLES
GOLDEN SHAPES: PENTAGRAM, GOLDEN TRIANGLE, ELLIPSE, & SPIRAL
FIBONACCI SEQUENCE & THE LIMITPATTERNS IN NATURE: FRACTALS &
CHAOS
“I used a square as the base shape. I did a tessellation by translation. The one side of the mobile is my tessellation, repeated on an angle. On the other side is a collage of tessellations and patterns. In the center there is a self-portrait of M.C.Esher. Most of the tessellations you see were done by him. I got the pictures from the Internet.”
FRACTALS KATARIN MACLEOD
Math Talk 2009
Fractal Basics• A rough or fragmented geometric shape• Exhibits self-similarity• First introduced in 1975 by Benoit
Mandelbrot• Term is derived from Latin (fractus)
meaning broken or fractured.• Based on a mathematics equation that
undergoes iterations whereby the equation is recursive
FRACTALS KATARIN MACLEOD
Math Talk 2009
Mendelbrot Fractal • Born November 20, 1924• Z Z2 + C, where c = a + bia) Fine structures at arbitrary small scalesb) To irregular to be use Euclidean geometryc) Usually has a Hausdorff dimension (greater than its topological dimension)d) Has a simple and recursive definition
FRACTALS KATARIN MACLEOD
Math Talk 2009
Koch Snowflake (1904)• Begin with an equilateral triangle and then
replace the middle of each third of every line segment with a pair of line segments that form an equilateral ‘bump’.
http://www.shodor.org/interactivate/activities/KochSnowflake/
FRACTALS KATARIN MACLEOD
Math Talk 2009
Sierpinski Triangle (1915)• Described by Polish mathematician
Waclaw Sierpinski.
• Is only self-similar therefore it is not a ‘true fractal’
http://www.arcytech.org/java/fractals/sierpinski.shtml
FRACTALS KATARIN MACLEOD
Math Talk 2009
Escape-time fractals• Known as ‘orbits’
• Defined formula or recurrence relation
• Examples: Mandelbrot set, Julia set, Burning ship fractal, Nova Fractal
FRACTALS KATARIN MACLEOD
Math Talk 2009
Iterative function systems• These have a fixed geometric replacement
rule – Koch snowflake, Sierpinski triangle
FRACTALS KATARIN MACLEOD
Math Talk 2009
Random Fractals• Generated by stochastic rather than
deterministic process
• Brownian motion, Levy flight, diffusion-limited aggregation
FRACTALS KATARIN MACLEOD
Math Talk 2009
Strange attractors• Generated by iteration of a map or solution
of a system of a system of initial valued differential equations that exhibit chaos.
http://www.fractal-vibes.com/fvc/Frame01.php3“Future Legends”
FRACTALS KATARIN MACLEOD
Math Talk 2009
References & resources
• http://www.pbs.org/wgbh/nova/fractals/program.html• http://en.wikipedia.org/wiki/Fractal• http://serendip.brynmawr.edu/playground/sierpinski.html• http://www.shodor.org/interactivate/activities/KochSnowflake/?
version=1.5.0_06&browser=MSIE&vendor=Sun_Microsystems_Inc.• http://www.geocities.com/CapeCanaveral/2854/• http://local.wasp.uwa.edu.au/~pbourke/fractals/burnship/• http://library.thinkquest.org/26242/full/types/ch14.html• http://www.ocf.berkeley.edu/~trose/rossler.html• http://groups.csail.mit.edu/mac/users/rauch/islands/• http://wapedia.mobi/en/L%C3%A9vy_flight• http://apricot.polyu.edu.hk/~lam/dla/dla.html• http://www.fractal-vibes.com/fvc/Frame01.php3
Math Talk 2009
L-System
• Aristid Lindenmayer (1925–1989). – Biologist and botanist.
• Studied the growth patterns of algae.
http://cage.rug.ac.be/~bh/L-systemen/Lindenmayer.htm
Math Talk 2009
L-System• L-systems were devised to provide a
mathematical description of the development of simple multi-cellular organisms, and to demonstrate relationships between plant cells.
• These systems are also used to describe higher plants and complex branching.
Math Talk 2009
Grammars• An alphabet is needed.• A set of fixed symbols known as
constants.• A initial word that starts everything. This is
called an axiom.• A set of production rules that describes
how the word is to be built.• Words are built iteratively, applying the
production rules at each iteration to form longer, more complex words.
Math Talk 2009
A More Complicated Example
• Alphabet: X, F
• Constants: +, -, [, ]
• Axiom: X
• Production rules:
X → F-[[X]+X]+F[+FX]-X
F → FF
Math Talk 2009
A More Complicated Example• Production rules:
X → F-[[X]+X]+F[+FX]-X
F → FF Steps:
0 X
1 F-[[X]+X]+F[+FX]-X
2 FF-[[F-[[X]+X]+F[+FX]-X]+F-[[X]+X]+F[+FX]- X]+FF[+FF F-[[X]+X]+F[+FX]-X]- F-[[X]+X]+F[+FX]-X
Math Talk 2009
What Does it Mean?• Suppose that we want to see “what the word
looks like”.• Now suppose we have one of these:
http://www.waynet.org/waynet/spotlight/2004/images/07/turtle640.jpg
Math Talk 2009
Turtle Graphics• F means “move forward”.
• + means “turn counterclockwise by a certain angle.”
• - means turn “clockwise by the same angle.”
http://www.terrapinswim.vicid.com/images/images/328/0/online_button.png
Math Talk 2009
Turtle Graphics (2)• [ means “remember location”.
• ] means “return to the point in memory”.
• X means “do nothing”. This is just a placeholder.
Math Talk 2009
Example 1
• Alphabet: F
• Constants: +, - (25)• Axiom: X
• Production rules:
X → F-[[X]+X]+F[+FX]-X
F → FF
Math Talk 2009
Example 1• Alphabet: F• Constants: +, - (25)• Axiom: X• Production rules:
X → F-[[X]+X]+F[+FX]-X
F → FF
Iteration 1546732
Math Talk 2009
Example 2
• Alphabet: F
• Constants: +, - (90)• Axiom: F-F-F-F
• Production rules:
F → FF-F-F-F-F-F+F
Math Talk 2009
Example 2• Alphabet: F• Constants: +, - (90)• Axiom: F-F-F-F• Production rules:
F → FF-F-F-F-F-F+F
Iteration 1432
Math Talk 2009
Example 3
• Alphabet: F
• Constants: +, - (25)• Axiom: F
• Production rules:
F → FF+[+F-F-F]-[-F+F+F]
Math Talk 2009
Example 3• Alphabet: F• Constants: +, - (25)• Axiom: F• Production rules:
F → FF+[+F-F-F]-[-F+F+F]
Iteration 154632
Math Talk 2009
Example 4
• Alphabet: F
• Constants: +, - (120)• Axiom: F+F+F
• Production rules:
F → F+F-F-F+F
Math Talk 2009
Example 4• Alphabet: F• Constants: +, - (120)• Axiom: F+F+F• Production rules:
F → F+F-F-F+F
Iteration 1546732
Math Talk 2009
http://www.royalhigh.edin.sch.uk/departments/departments/CDT/ahgc_0708_blender_vegetation.html
3D Trees Generated with an L-System
Math Talk 2009
“Hairy” Plants
Fuhrer, M.; Jensen, H.W.; Prusinkiewicz, P. “Modeling Hairy Plants”, Graphical Models 68(4), 333-342, 2006..
Math Talk 2009
Fractal Mountains
http://www.math.ucdavis.edu/~kapovich/fractal.gifhttp://myweb.cwpost.liu.edu/aburns/gallery/newgall.htm
6 February 2009 Jarvis MacLeod Wachowiak 62
Video Clips
Math/Art Videos: “Donald Duck In Mathmagicland” (1959) with host Donald Duck
“Life by the Numbers” (2006) with host Danny Glover